chemia - Studia

OMEGA AND SADHANA POLYNOMIALS IN P-TYPE SURFACE NETWORKS
If G is a co-graph then its orthogonal cuts C
1
, C2,...,
Ck
form a
partition of E(G): EG ( ) = C1∪C2
∪... ∪Ck, Ci ∩ Cj
=∅,
i≠ j.
A subgraph H ⊆ G is called isometric, if dH( u, v) = dG( u, v)
, for any
( uv , ) ∈ H; it is convex if any shortest path in G between vertices of H
belongs to H. The relation co is related to ~ (Djoković [21]) and Θ (Winkler [22])
relations [23,24].
Two edges e and f of a plane graph G are in relation opposite, e op
f, if they are opposite edges of an inner face of G. Then e co f holds by the
assumption that faces are isometric. The relation co is defined in the whole
graph while op is defined only in faces/rings. Note that John et al. [20]
implicitly used the “op” relation in defining the Cluj-Ilmenau index CI.
Relation op will partition the edges set of G into opposite edge strips
ops, as follows. (i) Any two subsequent edges of an ops are in op relation;
(ii) Any three subsequent edges of such a strip belong to adjacent faces;
(iii) In a plane graph, the inner dual of an ops is a path, an open or a closed
one (however, in 3D networks, the ring/face interchanging will provide ops which
are no more paths); (iv) The ops is taken as maximum possible, irrespective
of the starting edge. The choice about the maximum size of face/ring, and
the face/ring mode counting, will decide the length of the strip.
Also note that ops are qoc (quasi orthogonal cuts), meaning the
transitivity relation is, in general, not obeyed.
The Omega polynomial [25-27] Ω( x)
is defined on the ground of
opposite edge strips ops S1, S2,..., Sk
in the graph. Denoting by m, the number
of ops of cardinality/length s=|S|, then we can write
s
Ω ( x)
= ∑ m⋅x
(3)
s
On ops, another polynomial, called Sadhana Sd(x) is defined [28,29]:
| E( G)|
−s
Sd( x)
= ∑ m ⋅x
(4)
s
The first derivative (in x=1) can be taken as a graph invariant or a topological
index (e.g., Sd’(1) is the Sadhana index, defined by Khadikar et al. [30]):
Ω ′ (1) = ∑ m⋅ s = E ( G )
(5)
s
Sd′ (1) = ∑ m ⋅(| E( G) | −s)
(6)
s
An index, called Cluj-Ilmenau [20], CI(G), was defined on Ω ( x)
:
2
CI( G ) = {[ Ω′ (1)] −[ Ω ′(1) +Ω ′′(1)]}
(7)
In tree graphs, the Omega polynomial simply counts the non-opposite
edges, being included in the term of exponent s=1.
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